A practical method for calculating largest Lyapunov exponents from small
data sets

A practical method for
calculating largest Lyapunov exponents from small data sets

This article originally appeared in Physica D65:117-134,
1993. Please cite this publication when referencing this
material. Software that implements the algorithm described by this article
may be found here.

Abstract

Detecting the presence of chaos in a dynamical system is an important
problem that is solved by measuring the largest Lyapunov exponent.
Lyapunov exponents quantify the exponential divergence of initially close
state-space trajectories and estimate the amount of chaos in a system. We
present a new method for calculating the largest Lyapunov exponent from an
experimental time series. The method follows directly from the definition
of the largest Lyapunov exponent and is accurate because it takes
advantage of all the available data. We show that the algorithm is fast,
easy to implement, and robust to changes in the following quantities:
embedding dimension, size of data set, reconstruction delay, and noise
level. Furthermore, one may use the algorithm to calculate simultaneously
the correlation dimension. Thus, one sequence of computations will yield
an estimate of both the level of chaos and the system complexity.